transformations replaced by symmetry

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1: 19.15 Advantages of Symmetry

… ►Symmetry in $x,y,z$ of $R_F(x,y,z)$, $R_G(x,y,z)$, and $R_J(x,y,z,p)$ replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). … …

2: 19.22 Quadratic Transformations

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3: 19.25 Relations to Other Functions

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19.25.16 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=-\frac{1}{3}{\omega }^2{R}_J(c-1,c-k^2,c,c-{\omega }^2)+\sqrt{\frac{(c-1)(c-k^2)}{({\alpha }^2-1)(1-{\omega }^2)}}{R}_C(c({\alpha }^2-1)(1-{\omega }^2),({\alpha }^2-c)(c-{\omega }^2)),$$ ${\omega }^2=k^2/{\alpha }^2$.

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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4: Bille C. Carlson

… ►The main theme of Carlson’s mathematical research has been to expose previously hidden permutation symmetries that can eliminate a set of transformations and thereby replace many formulas by a few. …

5: Errata

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Just above (1.6.57) “clockwise” has been replaced with “anticlockwise”.

Suggested by Denys Bondar on 2024-04-10

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The text “greatest common divisor of $m,n$” was replaced with “greatest common divisor of $k,m$”.

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The reference Ismail (2005) has been replaced throughout by the further corrected paperback version Ismail (2009).

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This figure was rescaled, with symmetry lines added, to make evident the symmetry due to the inverse relationship between the two functions.

Reported 2015-11-12 by James W. Pitman.

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The entire Section was replaced.

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6: 18.7 Interrelations and Limit Relations

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§18.7(i) Linear Transformations

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§18.7(ii) Quadratic Transformations

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18.7.17 $$U_{2n}\left(x\right)=W_n\left(2x^2-1\right),$$

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Symbols:

$W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $U_n\left(x\right)$: Chebyshev polynomial of the second kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.30

Referenced by:

§18.7(ii), Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.7.E17

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_n\left(x\right)$, $W_n\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_n\left(x\right)$, $W_n\left(x\right)$, having been interchanged, on the right-hand side we replaced $V_n\left(2x^2-1\right)$, with $W_n\left(2x^2-1\right)$. For further details see Errata.

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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18.7.18 $$T_{2n+1}\left(x\right)=x{V}_n\left(2x^2-1\right).$$

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Symbols:

$T_n\left(x\right)$: Chebyshev polynomial of the first kind, $W_n\left(x\right)$: Chebyshev polynomial of the fourth kind, $V_n\left(x\right)$: Chebyshev polynomial of the third kind, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.5.29

Referenced by:

§18.7(ii), Erratum (V1.0.28) for Table 18.3.1

Permalink:

http://dlmf.nist.gov/18.7.E18

Encodings:

pMML, png, TeX

Correction (effective with 1.0.28):

The DLMF now adopts the definitions for the Chebyshev polynomials of the third and fourth kinds $V_n\left(x\right)$, $W_n\left(x\right)$ used in Mason and Handscomb (2003). Therefore $V_n\left(x\right)$, $W_n\left(x\right)$, having been interchanged, on the right-hand side we replaced $x{W}_n\left(2x^2-1\right)$, with $x{V}_n\left(2x^2-1\right)$. For further details see Errata.

See also:

Annotations for §18.7(ii), §18.7 and Ch.18

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7: 20.11 Generalizations and Analogs

… ►If both $m,n$ are positive, then $G(m,n)$ allows inversion of its arguments as a modular transformation (compare (23.15.3) and (23.15.4)): … ►

§20.11(v) Permutation Symmetry

… ►The importance of these combined theta functions is that sets of twelve equations for the theta functions often can be replaced by corresponding sets of three equations of the combined theta functions, plus permutation symmetry. …